Let $B_{\infty}(\Omega)$ be the space of bounded measurable functions on the measurable space $\Omega$. For a given Banach space $X$, let us denote $B_{\infty}(\Omega,X)$ by the set of all bounded measurable functions $f:\Omega\to X$ (meaning $f^{-1}(O)$ is measurable for every open $O$ in $X$).

Is the space $B_{\infty}(\Omega,X)$ just the invective tensor product of $X$ and $B_{\infty}(\Omega)$?